arxiv: 2605.13937 · v1 · submitted 2026-05-13 · 🌌 astro-ph.CO

Recognition: 2 theorem links

· Lean Theorem

Spherical collapse and cluster number counts in DHOST theories that pass the constraints from gravitational waves

Sakdithut Jitpienka , Khamphee Karwan , David F. Mota

Authors on Pith no claims yet

Pith reviewed 2026-05-15 02:35 UTC · model grok-4.3

classification 🌌 astro-ph.CO

keywords DHOST theoriesspherical collapsegalaxy cluster countsmodified gravitygravitational wave constraintseROSITA surveyΛCDM comparison

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The pith

DHOST theories consistent with gravitational wave constraints predict fewer high-redshift galaxy clusters than the standard ΛCDM model.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper studies spherical collapse and galaxy cluster abundance in a class of DHOST modified gravity theories that keep gravitational wave speed equal to light and avoid decay into scalars. Deviations from Einstein gravity grow significant near the end of matter domination, suppressing linear density growth on small scales while raising the critical overdensity needed for collapse. When cluster counts are computed with the analytic mass function and the mass threshold is fixed by matching ΛCDM predictions to eROSITA-inferred numbers, the DHOST models produce lower total counts, especially in the highest redshift bin. The reduction persists even when the present-day deviation is kept inside the tight bounds set by binary pulsar observations, although the authors note possible inaccuracies from extending the collapse model beyond general relativity.

Core claim

In DHOST theories where gravitational waves propagate at the speed of light without decaying into scalars, deviations from general relativity become appreciable at late times during the matter era. These deviations reduce the growth rate of linear matter perturbations on small scales while increasing the linearly extrapolated density contrast required for spherical collapse. As a result the analytic mass function yields lower galaxy cluster number counts than in ΛCDM; the suppression is strongest in the highest redshift bin even when the present-day deviation parameter is restricted to the upper limit allowed by binary pulsar data.

What carries the argument

Spherical collapse model adapted to DHOST, which supplies a modified collapse threshold δ_c that is inserted into the analytic halo mass function to compute redshift-dependent cluster number counts.

If this is right

Cluster abundance reaches its maximum in the lowest redshift bin.
The number of clusters in the highest redshift bin decreases as the deviation from Einstein gravity increases.
Even at the upper limit set by binary pulsar observations, DHOST cluster counts fall below the ΛCDM values calibrated to eROSITA.
The minimum mass used in the integration is chosen so that the ΛCDM prediction matches the eROSITA-inferred counts at each redshift.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Future high-redshift cluster surveys could tighten bounds on DHOST parameters beyond current pulsar limits if the analytic predictions hold.
The reported suppression may be testable with weak-lensing or Sunyaev-Zel'dovich cluster catalogs that reach similar redshifts.
Numerical N-body simulations in DHOST cosmologies would be required to check whether the spherical-collapse threshold remains a reliable proxy for halo formation.

Load-bearing premise

The spherical collapse model and analytic mass function remain accurate when applied to DHOST theories despite the change in gravitational dynamics.

What would settle it

A direct count of galaxy clusters in the highest redshift bin from eROSITA or an equivalent survey that equals or exceeds the ΛCDM prediction would falsify the predicted suppression for the allowed range of DHOST deviations.

Figures

Figures reproduced from arXiv: 2605.13937 by David F. Mota, Khamphee Karwan, Sakdithut Jitpienka.

**Figure 2.** Figure 2: FIG. 2: Evolution of [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: Relative difference in the growth factor, [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: Evolution of the extrapolated linear density contrast [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5: Number of galaxy clusters [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗

read the original abstract

We investigate the spherical collapse model and the abundance of galaxy clusters in a class of degenerate higher-order scalar--tensor (DHOST) theories in which gravitational waves do not decay into scalar perturbations and which are consistent with current constraints from gravitational-wave observations. We find that deviations from Einstein gravity can become significant at late times when the background universe is close to the scaling regime during the matter-dominated epoch. These deviations suppress the growth of linear matter perturbations on small scales while increasing the extrapolated linear density contrast at collapse, obtained from the spherical collapse model. Using the analytic mass function, we compute the corresponding cluster number counts. The minimum mass threshold in the mass integration for each redshift bin is determined by matching the predicted number counts in the $\Lambda$CDM model with those inferred from the eROSITA survey. We find that the cluster abundance reaches its maximum at low redshift bin, and that the number of clusters in the highest redshift bin is suppressed as the deviation from Einstein gravity becomes larger. The parameters of the theory are chosen such that the deviation from Einstein gravity at present is consistent with the local astrophysical bounds from binary pulsar observations. We find that even under such strict constraints, the upper bound on the deviation leads to lower predicted number counts compared with the $\Lambda$CDM model emulating the eROSITA survey results. However, this may be a consequence of the uncertainties in computing the number counts for the DHOST theories using the spherical collapse model and the analytical mass function.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper computes lower cluster counts in GW-constrained DHOST models via spherical collapse, but the offset rests on an unvalidated extension of the analytic machinery.

read the letter

The headline result is that even with DHOST parameters kept inside binary pulsar and gravitational wave bounds, the model gives fewer galaxy clusters than Lambda CDM once you solve the spherical collapse ODE in the modified background and insert the new threshold into the standard mass function. They calibrate the minimum mass by matching Lambda CDM predictions to eROSITA numbers, then show the DHOST case suppresses counts most clearly in the higher redshift bins. The deviation parameter is fixed externally so the present-day value respects local bounds, which keeps the exercise from being a fit to the clusters themselves. That part is straightforward and uses the external constraints cleanly. The work is a direct application of existing spherical collapse and analytic mass function techniques to this GW-safe subclass of DHOST theories, with the background evolution set so deviations appear at late times during matter domination. The calculation itself is explicit and the parameter choice avoids circularity. The soft spot is the one the abstract already names. The spherical collapse model and the Press-Schechter or Sheth-Tormen function are applied outside their original GR setting, and no N-body validation, excursion-set re-derivation, or direct test of the collapse threshold in DHOST is reported. The authors note that the reported suppression could be an artifact of those modeling uncertainties, so the quantitative difference from Lambda CDM remains conditional. No error bars or sensitivity checks on the ODE solution appear in the provided text either. This is for readers who already know cluster abundance forecasts in GR and want to see how a single DHOST deviation parameter shifts the counts under external constraints. Someone working on modified gravity tests with large-scale structure would get the most from it, but they would also see the need for better checks on the collapse dynamics. I would send it to peer review. The setup is concrete, the constraints are external, and the uncertainty is stated openly, so referees can focus on whether the analytic tools carry over.

Referee Report

2 major / 1 minor

Summary. The manuscript investigates spherical collapse and galaxy cluster number counts in DHOST theories consistent with gravitational wave and binary pulsar constraints. It reports that deviations from GR, even within allowed bounds, suppress linear perturbation growth on small scales, raise the collapse threshold, and result in lower cluster abundances than in ΛCDM when calibrated to eROSITA data, with the effect most pronounced at higher redshifts. The authors caution that the result may arise from uncertainties in extending the spherical collapse model and analytic mass functions to these theories.

Significance. If the extension of the spherical collapse model to DHOST is reliable, the work demonstrates that cluster surveys can probe deviations from GR that are permitted by local constraints, offering a potential observational test via suppressed number counts at high redshift. The use of external constraints to fix parameters and the matching to eROSITA results provide a clear, falsifiable prediction.

major comments (2)

[Abstract] Abstract: the headline result on suppressed cluster counts (relative to the eROSITA-matched ΛCDM case) is obtained by solving the spherical-collapse ODE for a modified δ_c(z) and inserting it into the standard Press-Schechter/Sheth-Tormen mass function; the abstract itself states that the observed suppression “may be a consequence of the uncertainties in computing the number counts for the DHOST theories using the spherical collapse model and the analytical mass function,” yet no N-body validation, excursion-set re-derivation, or direct comparison of the collapse threshold is reported.
[Cluster number counts section] Cluster number counts section: the minimum mass threshold for each redshift bin is fixed by matching the ΛCDM prediction to eROSITA-inferred counts, after which the DHOST-modified δ_c is used; no error propagation, sensitivity analysis to the deviation parameter, or robustness checks against variations in the mass-function fitting formula are provided, rendering the quantitative difference from ΛCDM conditional on an untested extrapolation.

minor comments (1)

[Results] The text should clarify whether the reported suppression is monotonic in the deviation parameter or exhibits any non-linear behavior at the upper bound allowed by binary-pulsar constraints.

Simulated Author's Rebuttal

2 responses · 2 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address each major comment point by point below, clarifying our methodology and the scope of the present work. Revisions have been made to strengthen the discussion of limitations and to include additional sensitivity checks.

read point-by-point responses

Referee: [Abstract] Abstract: the headline result on suppressed cluster counts (relative to the eROSITA-matched ΛCDM case) is obtained by solving the spherical-collapse ODE for a modified δ_c(z) and inserting it into the standard Press-Schechter/Sheth-Tormen mass function; the abstract itself states that the observed suppression “may be a consequence of the uncertainties in computing the number counts for the DHOST theories using the spherical collapse model and the analytical mass function,” yet no N-body validation, excursion-set re-derivation, or direct comparison of the collapse threshold is reported.

Authors: We agree that our results rely on the spherical collapse model to obtain a modified δ_c(z) and the standard analytic mass function. N-body simulations or a full excursion-set re-derivation for DHOST theories would provide stronger validation, but such computations are computationally demanding and lie outside the scope of this paper, which focuses on the analytic approach. In the revised manuscript we have added an explicit comparison of δ_c(z) between the DHOST models and ΛCDM, expanded the caveats section, and reinforced the cautionary language already present in the abstract regarding uncertainties in the extrapolation. revision: partial
Referee: [Cluster number counts section] Cluster number counts section: the minimum mass threshold for each redshift bin is fixed by matching the ΛCDM prediction to eROSITA-inferred counts, after which the DHOST-modified δ_c is used; no error propagation, sensitivity analysis to the deviation parameter, or robustness checks against variations in the mass-function fitting formula are provided, rendering the quantitative difference from ΛCDM conditional on an untested extrapolation.

Authors: We accept that the original submission lacked these quantitative checks. The revised version now includes a sensitivity analysis in which the DHOST deviation parameter is varied within the range allowed by binary-pulsar constraints, showing the resulting impact on cluster counts. We have also added a discussion of robustness with respect to the mass-function parameters used in the matching to eROSITA data and included a simple estimate of the uncertainty arising from that matching procedure. A complete propagation of all systematics and exhaustive variation of fitting formulas remains a more extensive project that we flag as future work. revision: yes

standing simulated objections not resolved

N-body validation or excursion-set re-derivation of the spherical collapse threshold and mass function in DHOST theories
Full systematic error propagation that incorporates all eROSITA observational uncertainties and exhaustive variations of mass-function fitting formulas

Circularity Check

0 steps flagged

No significant circularity; parameters externally constrained and cluster counts computed from independent ODE solution plus standard mass function

full rationale

The derivation sets theory parameters from external GW and binary-pulsar bounds, solves the spherical-collapse ODE in the DHOST background to obtain a new δ_c(z), inserts that threshold into the Press-Schechter/Sheth-Tormen mass function, and calibrates the minimum-mass cut by matching ΛCDM predictions to eROSITA counts. None of these steps reduces to a fitted quantity by construction, a self-definitional loop, or a load-bearing self-citation. The authors explicitly flag uncertainties in extending the analytic machinery, but this is an acknowledged limitation rather than circularity. The central quantitative claim (suppressed counts even at the pulsar-allowed upper bound) therefore remains a genuine prediction conditional on the validity of the spherical-collapse extension.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard cosmological background assumptions plus the domain assumption that spherical collapse and analytic mass functions transfer directly to DHOST; the deviation parameter is externally constrained rather than freely fitted.

free parameters (1)

deviation parameter from GR
Chosen so that present-day deviation remains consistent with binary-pulsar bounds; value not numerically specified in abstract.

axioms (2)

domain assumption Spherical collapse model applies to DHOST theories
Invoked to obtain the extrapolated linear density contrast at collapse.
domain assumption Analytic mass function remains valid for cluster abundance in modified gravity
Used to translate collapse threshold into number counts.

pith-pipeline@v0.9.0 · 5582 in / 1439 out tokens · 37405 ms · 2026-05-15T02:35:06.801747+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We solve Eq. (45) numerically... tune the initial value of δm such that δm → ∞ at a given redshift z... compute the corresponding linear density contrast at the collapse redshift, which defines δc.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The analytical mass function proposed by Press–Schechter... Sheth–Tormen... fS-T(σ) = A √(2a/π) [1+(σ²/aδc²)^p] δc/σ exp[-aδc²/2σ²]

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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